An online retailer determines that people click on a particular advertisement $2\%$ of the times it appears. Let $V$ be the number of times the advertisement appears to get the first person to click on it. Assume each click is independent. Find the probability that a person first clicks on the advertisement the $8^{\text{th}}$ time it appears. You may round your answer to the nearest hundredth. $P(V=8)=$
Answer: Without a fancy calculator For each appearance: $P({\text{click}})=0.02$ $P(\text{no click}})=0.98$ If the first person to click on the advertisement is the $8^{\text{th}}$ person to see it, the advertisement must appear $7$ times without a person clicking on it, followed by $1$ time when the a person does click. $\begin{aligned} P(V=8)&=P(\text{NNNNNNN}}{\text{C}}) \\\\ &=(0.98})(0.98})\cdots(0.98})({0.02}) \\\\ &=(0.98)^7(0.02) \\\\ &\approx0.01736 \end{aligned}$ $P(V=8) \approx 0.01736 \approx 0.02$